A Markov-type inequality for arbitrary plane continua

Markov’s inequality is sup [−1,1] |f ′| ≤ (deg f) sup [−1,1] |f |, for all polynomials f . We prove a precise version of this inequality with an arbitrary continuum in the complex plane instead of the interval [−1, 1]. Theorem 1. Let E be a continuum in the complex plane, and f a polynomial of degree d. Then capE sup E |f ′| ≤ 21/d−1d2 sup E |f |. (1) Here cap denotes the transfinite diameter (capacity) of a set [2, Ch. 2]. Using the well-known inequality diamE ≤ 4 capE we obtain Corollary. With the assumptions of Theorem 1 we have diamE sup E |f ′| ≤ 2d sup E |f |. (2) This inequality looks more elementary than (1) but we could not find a direct proof. ∗Supported by NSF grants DMS-0100512 and DMS-0244421.